Periodic-orbit theory of universal level correlations in quantum chaos

Abstract

Using Gutzwiller's semiclassical periodic-orbit theory we demonstrate universal behavior of the two-point correlator of the density of levels for quantum systems whose classical limit is fully chaotic. We go beyond previous work in establishing the full correlator such that its Fourier transform, the spectral form factor, is determined for all times, below and above the Heisenberg time. We cover dynamics with and without time reversal invariance (from the orthogonal and unitary symmetry classes). A key step in our reasoning is to sum the periodic-orbit expansion in terms of a matrix integral, like the one known from the sigma model of random-matrix theory.

Chaotic maps and flows: exact Riemann–Siegel lookalike for spectral fluctuations

“…is identical with the one previously obtained in [1,2,6]. In quite analogous a manner we recover, in our present context, the previous result Z −− ∼ 0.…”

“…Again, for the unitary symmetry class the diagonal approximation yields the exact finite-N CUE generating function and, with N → ∞, the exact GUE result. We recover the previously obtained pseudo-orbit expansions [1][2][3] for higher-order corrections in the limit N → ∞, for all Wigner-Dyson symmetry classes.…”

“…We Taylor expand all four pertinent exponentials and so obtain a fourfold sum over pseudo-orbits. Neglecting orbit repetitions and again representing the four phases ϕ A/B/C/D as in (17) we find [1,2,6]…”

“…Universal fluctuations in quantum energy spectra under conditions of full classical chaos are well understood in terms of Gutzwiller's semiclassical periodic-orbit theory [1,2]. The phenomenological description previously given by random-matrix theory has been fully recovered for individual (rather than ensemble averages of) systems from the unitary, orthogonal [1,2], and symplectic [3] symmetry classes.…”

“…The phenomenological description previously given by random-matrix theory has been fully recovered for individual (rather than ensemble averages of) systems from the unitary, orthogonal [1,2], and symplectic [3] symmetry classes.…”